Finite Group Databasebeta

← #44559849

Subgroups — #44559849

Order 32128.

480 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548747C212yesmaximal
2C4⋊Q8×D251C211yesmaximal, characteristic, complemented
3C4×#16860811C211yesmaximal, characteristic
4#30631897C211yesmaximal, characteristic, complemented
5#30631895C212yesmaximal, complemented
6#30631893C211yesmaximal, characteristic, complemented
7#30631891C212yesmaximal, complemented
8#30631889C211yesmaximal, characteristic, complemented
9#30631709C211yesmaximal, characteristic, complemented
10#30631694C212yesmaximal, complemented
11#462×C251C211yesmaximal, characteristic, complemented
12C2×#16860811C2211yescharacteristic
13C2×Q8×D251C2212yes—
14#22548747—28no—
15C2×#16860800C2212yes—
16C2×#16860799C2212yes—
17C2×#16860797C2212yes—
18C2×#16860793C2212yes—
19#22548715C2211yescharacteristic
20#22548714—21no—
21#22548714—21no—
22#22548713C2212yes—
23Q8×Dic251C2211yescharacteristic
24C4×D1004—21no—
25C4×D1004C2211yescharacteristic
26#22548677—21no—
27#22548677C2211yescharacteristic
28#22548673—21no—
29#22548673C2211yescharacteristic
30C4⋊C4×D251—21no—
31C4⋊C4×D251—21no—
32C42×D251—21no—
33C42×D251C2211yescharacteristic
34C4⋊(C251⋊C8)C2211yescharacteristic
35Dic251⋊C8C2212yes—
36#22548640C2212yes—
37#22548639C2212yes—
38#22548638C2212yes—
39C2×Q16×C251C2212yes—
40C4⋊Q8×C251C2211yescharacteristic
41C4×Q8×C251C2211yescharacteristic
42C4⋊C8×C251C2211yescharacteristic
43#103×C251C2212yes—
44#16860811—22no—
45#16860811D414yes—
46C2×Dic502—22no—
47C2×Dic502C2312yes—
48Q8×D251—24no—
49Q8×D251—24no—
50C2×D1004—21no—
51C2×D1004C2311yescharacteristic
52C2×C4×D251—21no—
53C2×C4×D251—22no—
54C2×C4×D251—21no—
55C2×C4×D251—21no—
56C2×C4×D251D411yescharacteristic
57C2×C4×D251D411yescharacteristic
58C2×C4×D251C2311yescharacteristic
59#16860800—24no—
60#16860799—24no—
61#16860797—24no—
62#16860793—24no—
63C2×C251⋊C8C2312yes—
64C4⋊Dic251—21no—
65C4⋊Dic251—21no—
66C4⋊Dic251—21no—
67C4⋊Dic251C2311yescharacteristic
68Dic251⋊C4—22no—
69Dic251⋊C4—22no—
70(C2×D251)⋊C4—21no—
71(C2×D251)⋊C4—21no—
72(C2×D251)⋊C4—21no—
73C4×Dic251—21no—
74C4×Dic251—21no—
75C4×Dic251D411yescharacteristic
76C4×Dic251D411yescharacteristic
77C4×Dic251C2311yescharacteristic
78C2×Q8×C251C2312yes—
79C2×Q8×C251C2311yescharacteristic
80Q16×C251—24no—
81C2×C8×C251C2312yes—
82C4⋊C4×C251—21no—
83C4⋊C4×C251—21no—
84C4⋊C4×C251—21no—
85C4⋊C4×C251C2311yescharacteristic
86C42×C251—21no—
87C42×C251C2311yescharacteristic
88C22×D251—21no—
89C22×D251C2×D411yescharacteristic
90Dic502—22no—
91Dic502—22no—
92Dic502—44no—
93D1004—24no—
94D1004—21no—
95D1004C2×D412yes—
96C2×Dic251—21no—
97C2×Dic251—22no—
98C2×Dic251C2×D411yescharacteristic
99C2×Dic251C2×D411yescharacteristic
100C2×Dic251C2×D411yescharacteristic
101C4×D251—22no—
102C4×D251—24no—
103C4×D251—22no—
104C4×D251—44no—
105C4×D251—42no—
106C4×D251—42no—
107C4×D251C2×D412yes—
108C4×D251#4112yes—
109C251⋊C8—22no—
110Q8×C251—21no—
111Q8×C251—22no—
112Q8×C251—22no—
113Q8×C251C2×D412yes—
114C2×C4×C251—21no—
115C2×C4×C251—21no—
116C2×C4×C251—22no—
117C2×C4×C251C2×D411yescharacteristic
118C2×C4×C251C2×D411yescharacteristic
119C2×C4×C251C2411yescharacteristic
120C8×C251—22no—
121C22×C251C22×D411yescharacteristic
122C2×D251—21no—
123C2×D251—21no—
124C2×D251—22no—
125C2×D251—41no—
126C2×D251—41no—
127C2×D251#12112yes—
128C4×C251—22no—
129C4×C251—21no—
130C4×C251—42no—
131C4×C251—41no—
132C4×C251#13712yescomplemented
133C4×C251C22×D411yescharacteristic
134C4×C251C2×#4111yescharacteristic
135Dic251—21no—
136Dic251—42no—
137Dic251—41no—
138Dic251#12112yes—
139Dic251#13712yescomplemented
140C2×C251C2×#12111yescharacteristic
141C2×C251C2×#13711yescharacteristic
142C2×C251C2×#13711yescharacteristic
143D251—22no—
144D251—42no—
145C251#303911yescharacteristic, complemented, Sylow
146#3039—2511nomaximal, Sylow
147C2×#137—2512no—
148C2×C4⋊Q8—2511no—
149C4×#41—2511no—
150#462—2512no—
151#462—2511no—
152#462D25111yescharacteristic, complemented
153#461—2511no—
154#461—2511no—
155#461—2512no—
156#423—2511no—
157#419—2512no—
158C2×#41—2511no—
159C22×Q8—2512no—
160#137—5028no—
161C2×Q16—2512no—
162C2×Q16C2×D25112yes—
163C2×SD16—2512no—
164C2×SD16—2512no—
165C2×M16—2512no—
166C4⋊Q8—5021no—
167C4⋊Q8—5021no—
168C4⋊Q8—2512no—
169C4⋊Q8—2511no—
170C4⋊Q8C2×D25111yescharacteristic
171C4×Q8—2511no—
172C4×Q8C2×D25111yescharacteristic
173C4×D4—5021no—
174C4×D4—5021no—
175C4×D4—2511no—
176C4×D4—2511no—
177#117—5021no—
178#117—2511no—
179C2×C4⋊C4—5021no—
180C2×C4⋊C4—5021no—
181C2×C42—5021no—
182C2×C42—2511no—
183C4⋊C8—2512no—
184C4⋊C8—2511no—
185C4⋊C8C2×D25111yescharacteristic
186#103—2512no—
187#103—2512no—
188#103—2512no—
189#103C2×D25112yes—
190#41—5022no—
191#41—2514no—
192C2×Q8—5022no—
193C2×Q8—5024no—
194C2×Q8—5024no—
195C2×Q8—2512no—
196C2×Q8C22×D25112yes—
197C2×Q8C22×D25111yescharacteristic
198C2×D4—5021no—
199C2×D4—2511no—
200C22×C4—5022no—
201C22×C4—5021no—
202C22×C4—5021no—
203C22×C4—5021no—
204C22×C4—2511no—
205C22×C4—2511no—
206C22×C4—2511no—
207Q16—24no—
208Q16—5024no—
209SD16—5024no—
210SD16—5024no—
211M16—5024no—
212C2×C8—2512no—
213C2×C8C22×D25112yes—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—5022no—
218C4⋊C4—5022no—
219C4⋊C4—5021no—
220C4⋊C4—5021no—
221C4⋊C4—5021no—
222C4⋊C4—2511no—
223C4⋊C4C22×D25111yescharacteristic
224C22⋊C4—5021no—
225C22⋊C4—5021no—
226C22⋊C4—5021no—
227C42—21no—
228C42—5021no—
229C42—5021no—
230C42—2511no—
231C42—2511no—
232C42—2511no—
233C42C22×D25111yescharacteristic
234C23—5021no—
235C23—2511no—
236Q8—21no—
237Q8—22no—
238Q8—22no—
239Q8—10044no—
240Q8—5022no—
241Q8—5022no—
242Q8D4×D25112yes—
243D4—5024no—
244D4—5021no—
245D4—2512no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—10044no—
250C2×C4—10042no—
251C2×C4—10042no—
252C2×C4—5024no—
253C2×C4—5022no—
254C2×C4—5021no—
255C2×C4—5022no—
256C2×C4—5022no—
257C2×C4—2511no—
258C2×C4—2511no—
259C2×C4—2511no—
260C2×C4—2512no—
261C2×C4—2512no—
262C2×C4D4×D25111yescharacteristic
263C2×C4D4×D25111yescharacteristic
264C2×C4C23×D25111yescharacteristic
265C8—22no—
266C8—5022no—
267C22—10041no—
268C22—10041no—
269C22—5022no—
270C22—5021no—
271C22—5021no—
272C22—2512no—
273C22C2×D4×D25111yescharacteristic
274C4—21no—
275C4—22no—
276C4—41no—
277C4—42no—
278C4—10041no—
279C4—10042no—
280C4—5021no—
281C4—2512no—
282C4—2512no—
283C4#2254874712yescomplemented
284C4C2×D4×D25111yescharacteristic
285C4#41×D25111yescharacteristic
286C2—10042no—
287C2—5022no—
288C2#121×D25111yescharacteristic
289C2#137×D25111yescharacteristic
290C2C2×#2254874711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#462×C251C2
socleC22×C251C22×D4
prime radicalC2×#16860811C22
hypercenter#462D251
hypocenterC251#3039
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C4⋊Q8×D251C2semidirect
4. #30631897C2semidirect
5. #30631895C2semidirect
6. #30631893C2semidirect
7. #30631891C2semidirect
8. #30631889C2semidirect
9. #30631709C2semidirect
10. #30631694C2semidirect
11. #462×C251C2semidirect
132. C4×C251#137semidirect
139. Dic251#137semidirect
145. C251#3039semidirect
152. #462D251semidirect
283. C4#22548747semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#225487472
C4⋊Q8×D2512
C4×#168608112
#306318972
#306318952
#306318932
#306318912
#306318892
#306317092
#306316942
#462×C2512
#3039251